How to prove that $\sum^{n}_{k=0}\begin{pmatrix} n \\ k \end{pmatrix}\frac{(-1)^{k}(2k)!!}{k!}=(-1)^{n}$?

Question

I'm trying to normalise a function and the normalisation is complete if I can prove that $$\sum^{n}_{k=0}\begin{pmatrix} n \\ k \end{pmatrix}\frac{(-1)^{k}(2k)!!}{k!}=(-1)^{n}$$ where $i!!$ is the double factorial, e.g if $k=3$, $(2k)!!=6!!=6\times 4\times 2$. Evaluating for the first few values of $n$ it is clear that this is true, however I haven't encountered a series like this before and therefore don't know where to start. Any ideas?